Note: all temperatures in the below are in fahrenheit.
Some produce is happier out of the refrigerator altogether.
Tomatoes, eggplant, winter squash, sweet potatoes, basil, and some others shouldn’t be stored below 55º; which is way way too warm for meat and milk. If you’ve got meat and/or milk in the fridge it should be below 40º; and most leafy greens, herbs (but not the basil!), cabbage family vegetables, and some others are happy that cold also – some of them will tolerate colder than others.
Storing basil too cold will turn it black and yucky, sweet potatoes and winter squash will rot a lot faster, eggplant will discolor and turn bitter, tomatoes as has been said will lose flavor. (You can refrigerate any of those after they’ve been cooked, though. At that point they’re no longer alive and the cold won’t affect them in the same way.) While just over 55º would be best (at least presuming the sweet potatoes and squash have been previously properly cured at higher temperatures), too warm is better than too cold: room temperature is much better than the fridge.
Garlic stores best with good air circulation. If it’s been stored cold for a long time, it won’t keep once it’s warmed up; but if it hasn’t been in cold storage, it’ll probably keep longest out of the refrigerator and somewhere with good air circulation. If you buy the stuff at a standard grocery store, though, it’s probably been held in cold storage quite a while and then shipped cold and then been put out on a shelf at warmer temperatures, and won’t keep very long no matter what you do.
Some things, such as peppers and summer squash, are actually happier somewhere inbetween 40º and 55º+; but there’s a limit to what it’s practical for most people to provide. Unless your house runs quite chilly, I’d put those in the refrigerator drawer.
The reason you get two refrigerator drawers is that some fruits give off ethylene gas, and some vegetables are sensitive to it and may have their flavor ruined. Other fruits and vegetables are neutral on the subject. You can look up a chart if you’re fussy; or you can just put fruit in one drawer and vegetables in the other, which will mostly work.